My philosophy is:

There are things in life we are not meant to understand.

 

The reason that the area diff is .5 as two of you have pointed out, and you are wondering how you get a differance of 1.0 squares right?

you have two shapes, divide 1 open square by the two shapes and you get .5 the differance. The diff is multiplied by two due to that in on the hypontenus angles down 1/2 in the other it is up 1/2, thefore making two. I could explain it a lot better in person or pointed to it, not good at putting things into words, I score high on all mathematical or science reasoning tests, poorly on any reading/writing test

if you do not believe me tell me, I will break out the whole mathematical crap.

 

Okay, understnad it has been a while sonce I have been to school, and my job does not, ever, require geometry, so here goes:

We will be using the area of 1 square (S) length (L) and heigth (H)
and hypotenuse (Y)
Triangles;
red H = 3
red L = 8
green H = 2
green L = 5

the other two shapes are of little to no importance

red Y = square root of L^2 + H^2
64 + 9 73
red Y = square root of 73
green Y = square root of 29

okay next we go over the areas.

area of all shapes how it appears (if slope of traingle was similar)
L x H /2 13x5 / 2 65/2 32.5 overall area = 32.5S

that is how it appears, with a straight line between lower left corner and upper right. but lets break it down to the two traingles and the other two shapes

red = LxH / 2 8x3 / 2 24/2 12S
green 5x2 / 2 10/2 5S
other two 3x5 15 15S
32S even ??????? wow whaddaya know? this is due to the uneven angles of the two triangles which has been talked about in previous posts.

there is a differance of .5 squares, so how do you get 1 whole open square in there?
Now just stay with me on this, I am not good with words.

The triangle formed by the straight line between the two points (overall Y) and the lines of the hypotenus of the green triangle and red triangles is equal two .5 squares. you can do the math yourself, I do not feel like doing the math for the square roots of 73 or 29. I do not need to to know this.
now being that this is equal to .5 squares, the angle formed by the two hypotenuses is below the 'overall Y' so you take AWAY the .5 area away from the area of the larger triangle that you think you see, which is 32.5 you get WOW! 32.
now to get this full 1 square, swap the red and green triangle positions around.the angle you have now created is ABOVE 'overall Y', so you have the same triangle using the two Y's from green and red, and the overall Y, just flipped and reversed. but still .5 squares. so you now have 1/2 a square above the overall Y 1/2 above, 1/2 below, there is your 1 full square.

Like I said, I am not the best with words. i hope this explains it.

It is amazing what a small differance in angles/rise/slope can make in the overall picture?

 

Space Time Contimunion

Wasting to much server Space

Wasting to much Truck Time

Wasting to much to Contimunion!!!

 

Here's how I explained it to my son using a simple analogy and fuzzy math.

Suppose that we're looking at a cutaway view of a roof we just built, the length of the roof from edge to center is 13 ft the height at the peak is 5 ft the roof is perfectly straight and we we are looking at a perfect right angle triangle. That would make the area under the roof (base times height divided by 2) 32.5 sq ft.

"Right?''....."Right."

OK now suppose a year later while looking closely at the roof you seem to notice that the weight of the roof has sagged it. So you go inside measure it with a string line and find the beam has stretched slightly and the center has sunken inward 6" (top figure) and this reduces the area under the roof by 1/2 a sq ft so the area under the roof is now 32 sq ft.

"Right?"..... "Umm I'll take your word for it." (Love fuzzy math)

OK now to fix it you get a jack and try to raise the center back to where it was but you over do it and raise it to much bowing the center of the roof outward 6" (bottom figure) Now because you made a mistake and raised the roof 6" more than it was when new you now have an area of 33 sq under the roof.

"Right? ..... "Right." (Fuzzy math strikes again)

OK so the area difference between the sunken roof (top figure) with an area of 32 sq ft and the roof you jacked up to much (bottom figure) with an area of 33 sq. ft is 1 sq ft.

"Right?" ..... "Wow! Right and the missing hole is 1 sq ft."

OK now for the hard part. Because the human brain wants to see any line that is very slighty bent or curved as a stright line you must convince your brain because that hole is there the roof line on those figures could not possibly be straight. Stare at it a couple minutes.

"Wow, Dad there different, I can see it." ... "So can I son."

Skip_T

Logic once more triumphs over screwed up human preception.

 

There is no need for drastic action here. She spent a couple of hours with it, then gave up. however, she did take it to work this morning, to totally mess with the entire think tank's schedule......

So, when she comes home tonight, she may have an answer, or she may be crestfallen, and a shadow of her former self, devastated by a problem too great for her feeble mind. Then I will have to cook dinner.

Thanks Alan, now look what you've done!

Theo

 

P.S. I apologize if my logic seems cunfusing I am not good with words, but it is there if you can sort through my confusing mind.


Theo, ask your wife if she can figure out what I was trying to say? And then see if she understands it.

 

She got me to print out your reply - the best brains in Memphis will be looking at it tomorrow, before their boss catches them and fires the lot of 'em for slacking.

She spent time on it today, but didn't figure it out..

Tomorrow is another day wasted at work....

Theo

Added as EDIT


One point. She drew out the picture again, on graph paper. It was with perfect angles and drawn accurately. It did the same as the posted picture, when cut up into shapes. She is baffled so far.....

Another Edit.

She is now having fun with it, and reckons that the triangles aren't actually the size you are led to think they are. The actual area apparently is 32½ squares, not the 32 or the 33 that the diagrams show.

She has totally lost me here, but that is not too hard.

Theo...

 

Okay, well get back to me after she gets a chance to study it, again I apologize if it is really as confusing as I think it is.

 

This is amazing - she is still hard at, with trigonometrical tables, and two calculators as well as her lap top.

PROGRESS REPORT:

the angle for a 13 x 5 triangle would be 21.037° at the point.
The other triangles are not the same, and their angle is 20.5° and 21.8° respectively..

Thereby looking identical but being totally different.

It is kind of cool that the difference is exactly one square..
The question I ask is Who thought it up??

She just checked eveything, said she did it right, and declared the above to be the explanation. I choose to believe her.

Theo

 

I think that making this problem would be alot easier than someone else trying to figure it out, especially being that you know the general idea of what you will do.

All you have to do is make a triangle so many squares by so many many squares, in this case 13x5.

Then make the decision that you want to in the bottom triangular shape that you want one extra square of area without it appearing so. how do you do this? you break up the hypotenuse in two eneven lines, if it wer even the fact taht it was would be easier to see upon initial inspection.

you raise the angle of these two lines so that the hypotenuse and these two loines create a triangle of .5 squares....at the angle formed by these two lines you draw 1 vertical line and one horizontal line, there by making your two triangles in the problem. you flip around the triangles as in the problem and VOILA!!! you have this scenario. You can do whatever you wish with the remaining rectangle between the two triangles as you see fit. They are almost useless in this problem except to simply take up space and to make you SEE the problem.





And when you said she agrees with the above explanation, where you reffering to mine?

 

Base,

You are a nut. I got it when you first explained it. Me and a buddy had noticed that the whole thing passed through different vertical points on the sloped line as it moved up from the left. When I read your reply I knew exactly what you were talking about. You are right. The other thing we noticed right away was that one triangle was a 2/5 and the other had a rise of 3 so it would have to be 7.5 wide to have the same angle. As you know it's 8 wide not 7.5. I can't see it right now or I'd say the colors but I'm sure you know what I'm talking about.

 

Thankyou, I am a nuit.

 

Here is the way I see it, only exaggerated, but still to scale in itself.

Note that there is now a difference of 14 units between the two figures that looked more like triangles in the first post.

 

the angle for a 13 x 5 triangle would be 21.037° at the point.
The other triangles are not the same, and their angle is 20.5° and 21.8° respectively..

Thereby looking identical but being totally different.

It is kind of cool that the difference is exactly one square..
The question I ask is Who thought it up??

She just checked eveything, said she did it right, and declared the above to be the explanation. I choose to believe her.

Her figuring is accurate, she got to it by following your explaination, as far as it went.

Theo